Aldo Rustichini, William G. Iacono, James J. Lee, and Matt McGue| Journal of Political Economy 2023 131:10, 2724-2779
.
└── Chromosome
└── DNA
└── Gene
└── Genome
Twin studies
Genome wide association studies (GWASs)
Regress outcome \(y^{i}_{j}\) on \(m\)-th SNP for all \(m=1, \dots, M\):
\[ y^{i}_{j} = \beta_{m}SNP^{i}_{jm}+\epsilon^{i}_{jm}, \quad m=1,\dots,M. \]
Remember:
Polygenic score = Predicted outcomes based on all relevant SNPs
\[ PGS^{i}_{j}=\sum_{m=1}^{M}\tilde{\beta}_{m} SNP^{i}_{jm} \]
flowchart LR
A1[father] --> A3(couple)
A2[mother] --> A3(couple)
A3[couple] --> B("child
genotype")
A3[couple] --"genetic nurture"--> C("child
environment")
B("child
genotype") --> D("child
education")
C("child
environment") <--"gene×environ"--> B("child
genotype")
C("child
environment") --> D("child
education")
Family \(i\) parents’ problem with twins \(j=1, 2\) of observable skills \(\theta^{i}_{j}\) \[ \begin{aligned} \max_{\{E^{i}, I^{i}_{1}, I^{i}_{2}\}} \;\;\;&\E\left[(1-\delta)\ln E^{i}+\delta\left(y^{i}_{1}+y^{i}_{2}\right)\left|\theta^{i}_{1}, \theta^{i}_{2}\right.\right]\\ \st \;\;\; & E^{i}+I^{i}_{1}+I^{i}_{2}=e^{y^{i}} \end{aligned} \tag{6} \] \[ \begin{align} \scriptsize{\mbox{Human capital production function}} && h^{i}_{j}&=\alpha_{I}\ln I^{i}_{j}+\alpha_{\theta}\theta^{i}_{j}+\epsilon^{\mathrm{h}i}_{j} && \phantom{mmmmmmmmmmmm} \tag{8}\label{eq8}\\ \scriptsize{\mbox{Child income}} && y^{i}_{j}&=\alpha_{\mathrm{h}}h^{i}_{j}+\epsilon^{\mathrm{y}i}_{j} &&\tag{9}\label{eq9} \end{align} \] Solution \[ I^{i}=\frac{\delta\alpha_{\mathrm{I}}\alpha_{\mathrm{h}}}{1-\delta+2\delta\alpha_{\mathrm{I}}\alpha_{\mathrm{h}}}e^{y^{i}} \tag{11} \] Optimised child income \[ y^{i}_{j}=\alpha_{\mathrm{I}}\alpha_{\mathrm{h}}\ln \frac{\delta\alpha_{\mathrm{I}}\alpha_{\mathrm{h}}}{1-\delta+2\delta\alpha_{\mathrm{I}}\alpha_{\mathrm{h}}} +\alpha_{\mathrm{I}}\alpha_{\mathrm{h}}y^{i}+\alpha_{\theta}\alpha_{\mathrm{h}}\theta^{i}_{j}+\alpha_{\mathrm{h}}\epsilon^{\mathrm{h}i}_{j}+\epsilon^{\mathrm{y}i}_{j} \tag{12}\label{eq12} \]
Matching of parents
Genetic transmission: \(\Delta\)(father genotype) × \(\Delta\)(mother genotype) \(\to\) \(\Delta\)(child genotype)
The appendix D of paper shows an invariant mapping exists with properties shown in Theorem 3.1
Skills production function (genotype enters here, hence the parental genotypes) \[ \theta^{i}_{j}=w\left(g^{i}_{j}\right)+\pi y^{i}+\bfPi\bfx^{i}_{j}+F^{i}+\epsilon^{\theta i}_{j} \tag{4}\label{eq4} \] Estimating equations: Substitute [4] into [12] gives child income decomposition, and [11] into [8] gives human capital decomposition \[ \begin{align} y^{i}_{j} &= \alpha_{\theta}\alpha_{\mathrm{h}}w\left(g^{i}_{j}\right) +\left(\alpha_{\mathrm{I}}\alpha_{\mathrm{h}}+\alpha_{\theta}\alpha_{\mathrm{h}}\pi\right)y^{i} +\alpha_{\theta}\alpha_{\mathrm{h}}F^{i}\\ &\hspace{1em} +\alpha_{\theta}\alpha_{\mathrm{h}}\bfPi\bfx^{i}_{j} +\alpha_{\theta}\alpha_{\mathrm{h}}\epsilon^{\theta i}_{j} +\alpha_{\mathrm{h}}\epsilon^{\mathrm{h} i}_{j} +\epsilon^{\mathrm{y}i}_{j}\tag{21}\label{eq21}\\ h^{i}_{j} &= \alpha_{\theta}w\left(g^{i}_{j}\right) +\left(\alpha_{\mathrm{I}}+\alpha_{\theta}\pi\right)y^{i} +\alpha_{\theta}F^{i}\\ &\hspace{1em} +\alpha_{\theta}\bfPi\bfx^{i}_{j} +\alpha_{\theta}\epsilon^{\theta i}_{j} +\epsilon^{\mathrm{y}i}_{j}\tag{23}\label{eq23} \end{align} \]
Can show IGE is smaller in “standard model” than genetic model
Becker and Tomes (1979) \[ \begin{align} \theta_{t+1} &= \eta \theta_{t}+\epsilon^{\theta}_{t+1}\tag{26}\label{eq26}\\ y^{i}_{t+1} &= \alpha_{\mathrm{I}}\alpha_{\mathrm{h}}y^{i}_{t}+\alpha_{\theta}\alpha_{\mathrm{h}}\theta_{t+1}+\epsilon^{\mathrm{y}i}_{t+1}\tag{25}\label{eq25} \end{align} \] Genetic model of this paper assumes \[ \begin{align} \theta &= w^{\theta}_{\mathrm{f}}\theta_{\mathrm{f}}+w^{\theta}_{\mathrm{m}}\theta_{\mathrm{m}}\tag{A4}\label{eqA4}\\ y^{i} &= w^{\mathrm{y}}_{\mathrm{f}}y^{i}_{\mathrm{f}}+w^{\mathrm{y}}_{\mathrm{m}}y^{i}_{\mathrm{m}}\tag{A3}\label{eqA3} \end{align} \] Plugging these give \[ \begin{align} \theta_{t+1} &= \eta \sum_{s=m,f}w^{\theta}_{s}\theta_{st}+\epsilon^{\theta}_{t+1}\tag{33}\label{eq33}\\ y^{i}_{t+1} &= \alpha_{\mathrm{I}}\alpha_{\mathrm{h}}\sum_{s=m,f}w^{\mathrm{y}}_{s}y^{i}_{st}+\alpha_{\theta}\alpha_{\mathrm{h}}\theta_{t+1}+\epsilon^{\mathrm{y}i}_{t+1}\tag{32}\label{eq32} \end{align} \]
Structural equation model (SEM)
Identification conditions of structural parameters remain the same as in simultaneous equation models
In this paper:
Controls used in estimation
Identification ideas
Identification ideas
\[ \begin{align} e^{i}_{\mathrm{h}} &= \alpha_{e_{\mathrm{h}}}+ \gamma^{\mathrm{pGS}_{m}}_{e_{\mathrm{h}}}\mathrm{pGS}^{i}_{m}+\gamma^{\mathrm{pGS}_{f}}_{e_{\mathrm{h}}}\mathrm{pGS}^{i}_{f}+\xi_{e_{\mathrm{h}}}\tag{43}\label{eq43}\\ y^{i}_{\mathrm{h}} &= \alpha_{y_{\mathrm{h}}}+\gamma^{\mathrm{pGS}_{m}}_{y_{\mathrm{h}}}\mathrm{pGS}^{i}_{m}+\gamma^{\mathrm{pGS}_{f}}_{y_{\mathrm{h}}}\mathrm{pGS}^{i}_{f}+\xi_{y_{\mathrm{h}}}\tag{44}\label{eq44}\\ e^{i}_{j} &= \alpha_{e}+\gamma^{e_{\mathrm{h}}}_{\mathrm{e}}e_{\mathrm{h}}^{i} +\gamma^{y_{\mathrm{h}}}_{\mathrm{e}}y_{\mathrm{h}}^{i} +\gamma^{\mathrm{pGS}}_{\mathrm{e}}\mathrm{pGS}^{i}_{j} +\gamma^{\mathrm{pGS}_{m}}_{\mathrm{e}}\mathrm{pGS}^{i}_{m} +\gamma^{\mathrm{pGS}_{f}}_{\mathrm{e}}\mathrm{pGS}^{i}_{f} +\xi_{\mathrm{e}}\tag{45}\label{eq45} \end{align} \]
Data: Minnesota Twin Study
Education years↑ with
Cognitive ability↑ with
Noncognitive ability↑ with
Dizygotic (fraternal) + monozygotic (identical) twins
Education years↑ with
Parental education years↑ with
Family income↑ with
感想